In a multistage rocket the upper stages are payload for the booster. They contribute to the mass ratio of the booster and thus affect it's performance. If the booster has a short burn duration it will be small, and the payload will drastically affect it's mass ratio. If it is large, it will have a long burn duration, and the payload will be a small fraction of it's mass ratio. So, what determines the stage ratio, how big should the booster be? Is there a magic number hidden somewhere in the equations?
The basic rocket equation is; Delta V = ISP * G * Ln(MR), and the guiding mantra of the Minimum Cost Design Method is; highest Delta V / Buck. This is one of those questions that requires a first year MBA student to "run the numbers", a spread sheet program would make this easy, IF I had one. The process is a tedious one of calculating the Delta V / $ for a booster payload combination, and then adding several different tank sizes to determine the cheapest solution. But, after a few notes, a few equations, several sheets of calculations, and several tables of results, I have uncovered several interesting facts.
* There is no economic benefit in long duration burn boosters. Dumping stages as early as possible is the cheapest way to get into space.
* Tankage and propellant cost have no bearing on determination of the staging ratio. Dumping stages as early as possible is the cheapest way to get into space.
* Engine costs and stage contract overhead are the only factors which favor long burn duration.
* Reusability removes the engine cost from the equation and favors short burn duration, not long burn duration. Dumping stages as early as possible is the cheapest way to get into space.
By making the stages as dumb as possible, with simple engines and by concentrating all the high technology into the top most stage, a multi stage approach has many benefits. This leads us to one very interesting conclusion; A Single Stage To Orbit - Reusable Launch Vehicle of the type NASA is spending BILLIONS OF DOLLARS to develop is exactly the wrong approach to low cost access to space. NASA's high cost high technology engines and huge contract overheads are the only justification for a SSTO - RLV. A reusable multistage launch vehicle with many small stages operated by a Free Enterprise Private Commercial Space Transportation Company is a far cheaper method of getting into space.
Two examples of proposed launch vehicles composed of many small stages are the German ORTAG, and the American Rocket Company modular booster designs.
THE ADOBE - THE ROCKET MADE OF MUD.
Interviewer - (to camera) Today we are on the farm of Lester Fud, the builder of the Adobe rocket. (to Mr. Fud) Mister Fud, you're not a rocket scientist, is that correct?
Mr. Fud - Yep, shore ain't no NATHA rocket scientist.
Interviewer - Yet you've designed and built this very large rocket, and you intend to launch tourists into space with it.
Mr. Fud - Yep, got hold of a secret government report on low cost launch vehicles. That NASA high technology stuff is just a sham.
Interviewer - I see you're rocket's made of adobe bricks and you're using cow dung and mesquite logs as propellants.
Mr. Fud - Yep, should make quite a stink when she lights up.
Interviewer - Wasn't it expensive building such a large rocket?
Mr. Fud - Nope, used non union Republicans and "foreign exchange students".
Interviewer - The Russians have offered to fly tourists into space for $20 million dollars, how much will your tourists pay?
Mr. Fud - $19.95, unless their real fat. I'll have to charge more for fat folks.
Interviewer - Who do you see as your main competition?
Mr. Fud - NASA, even though I'm just a hick farmer, I can still see that NASA can offer space rides for free.